Those are probably female plants. Were you growing an all-male type?

If it was me....I would leave them, they will add Asparagus to your bed (you can't have too much Asparagus ;-)
The Males are better (but mainly to a commercial grower ,whose livelihood is based upon critical percentages ,their bottom line has to be met or they don't make a profit), but for a backyard gardener, the difference in production is maybe 10% less than optimum. Totally Wonderful... Enjoy

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I picked some and put it in a pan within 2 hours of picking. That was some good eating. Olive oil, garlic, onion and some course black pepper lightly sauteed. Good stuff!

The "small fern looking plants" coming up in your asparagus bed are probably asparagus volunteers as you suspected. Seedlings are not much larger than a pencil lead the first year and get ferny leaved right away. It will take a couple of years or more for them to grow to usable size. As Jon says, leave them and they will increase your crop.

Jim

Wow, jon.
Those are huge....are some literally as thick as a quarter??

What year are they? (how old) and What variety?

They are awesome, the last pic is my sixth harvest this year...so far ;-)
I planted the crowns last year 2-18-2010

I have no idea what they are, but they sure are delicious ;-)

Here is a link that might be useful: French Intensive/BioDynamic Asparagus (see all four)

amazing!
Only a year old, too?
Were the stems thick when they came up last year? and I'm assuming you didn't harvest anything last year....
how many plants and how often are you harvesting?

I didn't take any last year, 16 plants and I planted more this year ;-)

They were all thin and went right to "fern"....

Every couple of days ;-)

I just noticed some new little ferny shoots too. I will cover with some soil and then cover with straw in the fall